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Full text of "Mathematical And Physical Papers - Iii"

To eliminate el and e2 from (3), we may observe that we have very nearly
c'    el        ct + *'
and similar expressions hold good for e'} &c.    Hence
-K = S----72        / \    j.     1   +   Al    ,    4.   ~~*
We might, if required, express ct and c2 in terms of c by the formulse of common optics, without making any supposition as to the magnitude of t. In practice, however, t is usually small compared with c, cjt &c., so that we may simplify the above expression by retaining only the first power of t. We thus get
4. Before proceeding to apply this expression, let us investigate the conditions of distinctness. Denoting by A>rZ, <\VR the additions to R on account of the terms involving &\ y, we get from (1)
A T>    i A     l \   o    fa'    a'\
AB=/+/flf     +   ^
l /I     1
and Ay-R may be obtained by interchanging a and 6, # and We have by the formute of common optics
c.        r       c     c9 + t    s     c, + t'   c,        r        c '     ^ '
i                                                        A                                      1    '                      '2                                         ,'J
whence, supposing t small, expanding as far as the first power of t, and putting for shortness
so that p is the radius of a speculum having the same focal length as the mirror, we obtain